1/1*2+1/2*3+1/3*4+...+1/2007*2008+1/2008*2009
=(1-1/2)+(1/2-1/3)+(1/3-1/4)+...+(1/2007-1/2008)+(1/2008-1/2009)
=1-1/2009
=2008/2009.
1/1*2+1/2*3+1/3*4+...+1/2007*2008+1/2008*2009
=(1-1/2)+(1/2-1/3)+(1/3-1/4)+...+(1/2007-1/2008)+(1/2008-1/2009)
=1-1/2009
=2008/2009.